Bibliographic record
Abstract
This article is concerned with the relation between several classical and well-known objects: triangle Fuchsian groups, \mathbb{C}^\times -equivariant singularities of plane curves, torus knot complements in the 3-sphere. The prototypical example is the modular group PSL_2(\mathbb Z) : the quotient of the nonzero tangent bundle on the upper-half plane by the action of PSL_2(\mathbb Z) is biholomorphic to the complement of the plane curve z^3-27w^2=0 . This can be shown using the fact that the algebra of modular forms is doubly generated, by g_2,g_3 , and the cusp form \Delta=g_2^3-27g_3^2 does not vanish on the half-plane. As a byproduct, one finds a diffeomorphism between PSL_2(\mathbb R)/PSL_2(\mathbb Z) and the complement of the trefoil knot - the local knot of the singular curve. This construction is generalized to include all (p,q,\infty) -triangle groups and, respectively, curves of the form z^q+w^p=0 and (p,q) -torus knots, for p,q co-prime. The general case requires the use of automorphic forms on the simply connected group \widetilde{SL_2}(\mathbb R) . The proof uses ideas of Milnor and Dolgachev, which they introduced in their studies of the spectra of the algebras of automorphic forms of cocompact triangle groups (and, more generally, uniform lattices). It turns out that the same approach, with some modifications, allows one to handle the cuspidal case.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; both teacher heads agree on what is shown here.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".